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An alternating direction method of multipliers for tensor complementarity problems

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Abstract

The tensor complementarity problem (TCP) is a special instance of nonlinear complementarity problems, which has many applications in multi-person noncooperative games, hypergraph clustering problems, and traffic equilibrium problems. How to solve the TCP, via analyzing the structure of the related tensor, is one of important research issues. In this paper, we propose an alternating direction method of multipliers (ADMM) to solve the TCP. We show that the solution set of the TCP, where the involved multilinear mapping is monotone, is nonempty and compact if the involved tensor is an S-tensor. Moreover, the ADMM for the TCP with a monotone involved multilinear mapping is proven to be globally convergent with a linear convergence rate. Some preliminary numerical results show that the proposed ADMM method is promising and effective.

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References

  • Bai X, Huang ZH, Qi L (2016) Global uniqueness and solvability for tensor complementarity problems. J Optim Theory Appl 170:72–84

    Article  MathSciNet  Google Scholar 

  • Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci 2:183–202

    Article  MathSciNet  Google Scholar 

  • Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 3:1–122

    Article  Google Scholar 

  • Bruckstein AM, Donoho DL, Elad M (2009) From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev 51:34–81

    Article  MathSciNet  Google Scholar 

  • Che M, Qi L, Wei Y (2016) Positive-definite tensors to nonlinear complementarity problems. J Optim Theory Appl 168:475–487

    Article  MathSciNet  Google Scholar 

  • Che M, Qi L, Wei Y (2019) Stochastic \(R_0\) tensors to stochastic tensor complementarity problems. Optim Lett 13:261–279

    Article  MathSciNet  Google Scholar 

  • Chen C, Zhang LP (2019) Finding Nash equilibrium for a class of multi-person noncooperative games via solving tensor complementarity problem. Appl Numer Math 145:458–468

    Article  MathSciNet  Google Scholar 

  • Cottle RW, Pang JS, Stone RE (1992) The linear complementarity problem. Academic Press, Boston

    MATH  Google Scholar 

  • Douglas J, Rachford HH (1956) On the numerical solution of the heat conduction problem in 2 and 3 space variables. Trans Am Math Soc 82:421–439

    Article  MathSciNet  Google Scholar 

  • Du SQ, Zhang LP (2019) A mixed integer programming approach to the tensor complementarity problem. J Global Optim 73:789–800

    Article  MathSciNet  Google Scholar 

  • Du S, Che M, Wei Y (2020a) Stochastic structured tensors to stochastic complementarity problems. Comput Optim Appl 75:649–668

  • Du S, Ding W, Wei Y (2020b) Acceptable solutions and backward errors for tensor complementarity problems. J Optim Theory Appl. https://doi.org/10.1007/s10957-020-01774-y

  • Eckstein J, Bertsekas DP (1992) On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math Program 55:293–318

    Article  MathSciNet  Google Scholar 

  • Facchinei F, Pang JS (2003) Finite-dimensional variational inequalities and complementarity problems, vols I and II. Springer, New York

  • Gabay D, Mercier B (1976) A dual algorithm for the solution of nonlinear variational problems via finite-element approximations. Comput Math Appl 2:17–40

    Article  Google Scholar 

  • Goldfarb D, Ma S (2012) Fast multiple-splitting algorithms for convex optimization. SIAM J Optim 22:533–556

    Article  MathSciNet  Google Scholar 

  • Goldfarb D, Ma S, Scheinberg K (2013) Fast alternating linearization methods for minimizing the sum of two convex functions. Math Program 141:349–382

    Article  MathSciNet  Google Scholar 

  • Guo Q, Zheng MM, Huang ZH (2019) Properties of S-tensor. Linear Multi Algebra 67:685–696

    Article  MathSciNet  Google Scholar 

  • Han LX (2019) A continuation method for tensor complementarity problems. J Optim Theory Appl 180:949–963

    Article  MathSciNet  Google Scholar 

  • Han J, Xiu N, Qi H (2006) Nonlinear complementarity theory and algorithms. Shanghai Science and Technology Press, Shanghai (in Chinese)

  • Harker PT, Pang JS (1990) Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math Program 48:161–220

    Article  MathSciNet  Google Scholar 

  • He B, Yuan X (2012) On the \(O(1/n)\) convergence rate of the Douglas–Rachford alternating direction method. SIAM J Numer Anal 50:700–709

    Article  MathSciNet  Google Scholar 

  • He B, Tao M, Yuan X (2012) Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J Optim 22:313–340

    Article  MathSciNet  Google Scholar 

  • Hong M, Luo ZQ (2017) On the linear convergence of the alternating direction method of multipliers. Math Program 162:165–199

    Article  MathSciNet  Google Scholar 

  • Huang ZH, Qi L (2017) Formulating an \(n\)-person noncooperative game as a tensor complementarity problem. Comput Optim Appl 66:557–576

    Article  MathSciNet  Google Scholar 

  • Huang ZH, Qi L (2019a) Tensor complementarity problems—part I: basic theory. J Optim Theory Appl 183:1–23

  • Huang ZH, Qi L (2019b) Tensor complementarity problems—part III: applications. J Optim Theory Appl 183:771–791

  • Lin T, Ma S, Zhang S (2015) On the global linear convergence of the ADMM with multi-block variables. SIAM J Optim 25:1478–1497

    Article  MathSciNet  Google Scholar 

  • Ming ZY, Zhang LP, Qi L (2020) Expected residual minimization method for monotone stochastic tensor complementarity problem. Comput Optim Appl 77:871–896

    Article  MathSciNet  Google Scholar 

  • Ni Q, Qi L (2015) A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map. J Glob Optim 61:627–641

    Article  MathSciNet  Google Scholar 

  • Qi L (2005) Eigenvalues of a real supersymmetric tensor. J Symb Comput 40:1302–1324

    Article  MathSciNet  Google Scholar 

  • Qi L (2013) Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl 439:228–238

    Article  MathSciNet  Google Scholar 

  • Qi L, Huang ZH (2019) Tensor complementarity problems—part II: solution methods. J Optim Theory Appl 183:365–385

    Article  MathSciNet  Google Scholar 

  • Qi L, Luo Z (2017) Tensor analysis: spectral theory and special tensors. SIAM, Philadelphia

    Book  Google Scholar 

  • Qi L, Chen H, Chen Y (2018) Tensor eigenvalues and their applications. Springer, Singapore

    Book  Google Scholar 

  • Song Y, Qi L (2015) Properties of some classes of structured tensors. J Optim Theory Appl 165:854–873

    Article  MathSciNet  Google Scholar 

  • Song Y, Qi L (2017) Properties of tensor complementarity problem and some classes of structured tensors. Ann Appl Math 33:308–323

    MathSciNet  MATH  Google Scholar 

  • Song Y, Yu G (2016) Properties of solution set of tensor complementarity problem. J Optim Theory Appl 170:85–96

    Article  MathSciNet  Google Scholar 

  • Sun D, Toh KC, Yang L (2015) A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints. SIAM J Optim 25:882–915

    Article  MathSciNet  Google Scholar 

  • Tao M, Yuan X (2011) Recovering low-rank and sparse components of matrices from incomplete and noisy observations. SIAM J Optim 21:57–81

    Article  MathSciNet  Google Scholar 

  • Wang Y, Huang ZH, Qi L (2018) Global uniqueness and solvability of tensor variational inequalities. J Optim Theory Appl 177:137–152

    Article  MathSciNet  Google Scholar 

  • Wang X, Che M, Qi L, Wei Y (2020) Modified gradient dynamic approach to the tensor complementarity problem. Optim Methods Softw 35:394–415

    Article  MathSciNet  Google Scholar 

  • Xie SL, Li DH, Xu HR (2017) An iterative method for finding the least solution to the tensor complementarity problem. J Optim Theory Appl 175:119–136

    Article  MathSciNet  Google Scholar 

  • Zhang LP, Qi L, Zhou G (2014) M-tensors and some applications. SIAM J Matrix Anal Appl 35:437–452

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Haoran Zhu & Liping Zhang

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  1. Haoran Zhu
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  2. Liping Zhang
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Correspondence to Liping Zhang.

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Communicated by Jinyun Yuan.

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This work was supported by the National Natural Science Foundation of China (Grant No. 11771244).

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Zhu, H., Zhang, L. An alternating direction method of multipliers for tensor complementarity problems. Comp. Appl. Math. 40, 106 (2021). https://doi.org/10.1007/s40314-021-01499-2

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  • DOI: https://doi.org/10.1007/s40314-021-01499-2

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